Shape complexity and fractality of fracture surfaces of swelled isotactic polypropylene with supercritical carbon dioxide

We have investigated the fractal characteristics and shape complexity of the fracture surfaces of swelled isotactic polypropylene Y1600 in supercritical carbon dioxide fluid through the consideration of the statistics of the islands in binary SEM images. The distributions of area $A$, perimeter $L$, and shape complexity $C$ follow power laws $p(A)\sim A^{-(\mu_A+1)}$, $p(L)\sim L^{-(\mu_L+1)}$, and $p(C)\sim C^{-(\nu+1)}$, with the scaling ranges spanning over two decades. The perimeter and shape complexity scale respectively as $L\sim A^{D/2}$ and $C\sim A^q$ in two scaling regions delimited by $A\approx 10^3$. The fractal dimension and shape complexity increase when the temperature decreases. In addition, the relationships among different power-law scaling exponents $\mu_A$, $\mu_B$, $\nu$, $D$, and $q$ have been derived analytically, assuming that $A$, $L$, and $C$ follow power-law distributions.